Leonard Scott (University of Virginia)
Friday 3rd November, 12.05-12.55pm, Carslaw 373
Reduced standard modules and 1-cohomology
(This is joint work with Ed Cline and Brian Parshall.)
First cohomology groups of finite groups with nontrivial irreducible coefficients have been useful in several geometric and arithmetic contexts, including Wiles's famous paper on modularity of elliptic curves. Internal to group theory, 1-cohomology plays a role in the general theory of maximal subgroups of finite groups, as developed by Aschbacher and Scott.
One can easily pass to the case where the group acts faithfully, and the underlying module is absolutely irreducible. In this case, R. Guralnick conjectured that there is a universal constant bounding all of the dimensions of these cohomology groups. The work described in this talk provides the first general positive results on this conjecture, proving that the generic 1-cohomology H1gen(G,L):=limq H1(G(q),L) of a finite group G(q) of Lie type, with absolutely irreducible coefficients L, is bounded by a constant depending only on the root system.
This result emerges as a consequence of a general study, of interest in its own right, of the homological properties of certain rational modules Delta(lambda), Nabla(lambda), indexed by dominant weights lambda, for a reductive group G. The modules Delta(lambda) and Nabla(lambda) arise naturally from irreducible representations of the quantum enveloping algebra Uzeta (of the same type as G) at a pth root of unity, where p>0 is the characteristic of the defining field for G. When the Lusztig character formula holds for irreducible G-modules having restricted high weight, the modules Delta(lambda) and Nabla(lambda), for all regular dominant weights lambda, have very strong homological properties. These homological properties then provide a way to determine the desired bounds on generic 1-cohomology. The investigation leads to numerous related results and conjectures, involving, for example, structural relations between the Delta-modules and classical Weyl modules.